Extreme and Singular Behavior in Fundamental Models of Fluid Mechanics

这个数学研究项目将开发和应用理论和计算工具，以系统地搜索流体力学的一些基本方程的解的极端行为。目标是从第一原理中推导出物理上重要的量的精确预测，并了解潜在的非线性动力学机制。这项工作利用了首席研究员，合作者和其他科学家的最新发展，以实现最优控制理论和变分法的思想，以找到实现最大混合，最佳运输或其他极端耗散的流动。输运、混合和耗散是流体流动的最基本特征之一，它们对于从微流体工程到气候科学和天体物理学的重要应用具有基础性意义。这里采用的控制和优化技术构成了新的计算辅助分析方法，这些问题。该项目直接涉及到主要研究者下的研究生和博士后研究人员的高级培训？主要研究流体力学中的偏微分方程（包括Navier-Stokes方程和对流扩散方程）解的定性和定量性质。研究人员与密歇根大学的研究生和博士后研究人员以及其他地方的其他研究人员合作，采用现代应用分析和科学计算。主要包括四个方面的工作：（1）研究对流扩散方程解的定性和定量性质，并根据搅拌流的特点确定混合的极限，最终目的是了解湍流的混合效果;（二）利用新的优化方案来研究非线性动力系统中的极端时间平均值，并研究它们对经典模型的影响瑞利-B nard对流;（3）在三维Navier-Stokes方程中，采用计算最优控制方法寻找最大涡度拟能产生;以及（4）将分析和计算技术应用于来自生物学中的跨学科应用的其它非线性和/或随机动力学模型，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

This mathematical research project will develop and apply theoretical and computational tools to systematically search for extreme behavior in solutions of some of the fundamental equations of fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles and understand the underlying nonlinear dynamical mechanisms. The work capitalizes on recent developments of the Principal Investigator, collaborators, and other scientists to implement ideas from optimal control theory and the calculus of variations to find flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and they are of foundational significance for important applications ranging from microfluidics engineering to climate science and astrophysics. The control and optimization techniques adopted here constitute novel computationally aided analysis approaches to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers under the Principal Investigator?s direction at the University of Michigan.The research is centered on studies of qualitative and quantitative properties of solutions of partial differential equations in fluid mechanics including the Navier-Stokes and advection-diffusion equations. The investigator employs modern applied analysis and scientific computation in collaboration with graduate students and postdoctoral researchers at the University of Michigan, and with other researchers elsewhere. The four major components of the work are: (1) to study qualitative and quantitative properties of solutions to advection-diffusion equations and determine limits on mixing in terms of features of the stirring flows with an ultimate goal of understanding mixing effectiveness of turbulence; (2) to utilize new optimization schemes to investigate extreme time averages in nonlinear dynamical systems and investigate their implications for classical models of Rayleigh-B?nard convection; (3) to implement computational optimal control methods to search for maximal enstrophy production in the three-dimensional Navier-Stokes equations; and (4) to apply the analysis and computation techniques to other nonlinear and/or stochastic dynamical models from interdisciplinary applications in biology, chemistry and physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.